Math / Algebra

QuestionQuestion 1-7
Use the selected values of a linear function $g(x)$ in the table and the equation $h(x)$ shown below to evaluate $(g \circ h)(-3)$ . \begin{tabular}{|c|c|} \hline $x$ & $g(x)$ \\ \hline-3 & -5 \\ \hline-1 & -1 \\ \hline 0 & 1 \\ \hline 4 & 9 \\ \hline \end{tabular} $h(x)=x^{2}-2$ $-22$

Studdy Solution

STEP 1

1. We are given a linear function $g(x)$ with selected values in a table.
2. We have the function $h(x) = x^2 - 2$ .
3. We need to evaluate the composite function $(g \circ h)(-3)$ .

STEP 2

1. Evaluate $h(-3)$ using the function $h(x)$ .
2. Use the result from step 1 to find $g(h(-3))$ using the table of values for $g(x)$ .

STEP 3

Substitute $x = -3$ into $h(x)$ :
$h(-3) = (-3)^2 - 2$

STEP 4

Simplify the expression:
$h(-3) = 9 - 2$ $h(-3) = 7$

STEP 5

Now, we need to evaluate $g(h(-3)) = g(7)$ .
Check the table for the value of $g(7)$ .

STEP 6

Since $g(x)$ is a linear function and the value for $g(7)$ is not directly given in the table, we need to find the equation of the line using the given points.
Use two points from the table to find the slope $m$ :
Points: $(-3, -5)$ and $(4, 9)$
$m = \frac{9 - (-5)}{4 - (-3)} = \frac{9 + 5}{4 + 3} = \frac{14}{7} = 2$

STEP 7

Now, use the point-slope form to find the equation of the line:
Using point $(-3, -5)$ :
$y + 5 = 2(x + 3)$
Simplify to find $g(x)$ :
$y = 2x + 6 - 5$ $y = 2x + 1$
Thus, $g(x) = 2x + 1$ .

STEP 8

Now substitute $x = 7$ into $g(x)$ :
$g(7) = 2(7) + 1$ $g(7) = 14 + 1$ $g(7) = 15$
The value of $(g \circ h)(-3)$ is:
$\boxed{15}$

Was this helpful?

Studdy Solution

STEP 1

1. We are given a linear function $g(x)$ with selected values in a table.
2. We have the function $h(x) = x^2 - 2$ .
3. We need to evaluate the composite function $(g \circ h)(-3)$ .

STEP 2

1. Evaluate $h(-3)$ using the function $h(x)$ .
2. Use the result from step 1 to find $g(h(-3))$ using the table of values for $g(x)$ .

STEP 3

Substitute $x = -3$ into $h(x)$ :
$h(-3) = (-3)^2 - 2$

STEP 4

Simplify the expression:
$h(-3) = 9 - 2$ $h(-3) = 7$

STEP 5

Now, we need to evaluate $g(h(-3)) = g(7)$ .
Check the table for the value of $g(7)$ .

STEP 6

STEP 7

Now, use the point-slope form to find the equation of the line:
Using point $(-3, -5)$ :
$y + 5 = 2(x + 3)$
Simplify to find $g(x)$ :
$y = 2x + 6 - 5$ $y = 2x + 1$
Thus, $g(x) = 2x + 1$ .

STEP 8

Now substitute $x = 7$ into $g(x)$ :
$g(7) = 2(7) + 1$ $g(7) = 14 + 1$ $g(7) = 15$
The value of $(g \circ h)(-3)$ is:
$\boxed{15}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. We are given a linear function g(x) g(x) g(x) with selected values in a table.2. We have the function h(x)=x2−2 h(x) = x^2 - 2 h(x)=x2−2.3. We need to evaluate the composite function (g∘h)(−3) (g \circ h)(-3) (g∘h)(−3).

STEP 2

1. Evaluate h(−3) h(-3) h(−3) using the function h(x) h(x) h(x).2. Use the result from step 1 to find g(h(−3)) g(h(-3)) g(h(−3)) using the table of values for g(x) g(x) g(x).

STEP 3

Substitute x=−3 x = -3 x=−3 into h(x) h(x) h(x):h(−3)=(−3)2−2 h(-3) = (-3)^2 - 2 h(−3)=(−3)2−2

STEP 4

Simplify the expression:h(−3)=9−2 h(-3) = 9 - 2 h(−3)=9−2 h(−3)=7 h(-3) = 7 h(−3)=7

STEP 5

Now, we need to evaluate g(h(−3))=g(7) g(h(-3)) = g(7) g(h(−3))=g(7).Check the table for the value of g(7) g(7) g(7).

STEP 6

STEP 7

Now, use the point-slope form to find the equation of the line:Using point (−3,−5) (-3, -5) (−3,−5):y+5=2(x+3) y + 5 = 2(x + 3) y+5=2(x+3)Simplify to find g(x) g(x) g(x):y=2x+6−5 y = 2x + 6 - 5 y=2x+6−5 y=2x+1 y = 2x + 1 y=2x+1Thus, g(x)=2x+1 g(x) = 2x + 1 g(x)=2x+1.

STEP 8

Now substitute x=7 x = 7 x=7 into g(x) g(x) g(x):g(7)=2(7)+1 g(7) = 2(7) + 1 g(7)=2(7)+1 g(7)=14+1 g(7) = 14 + 1 g(7)=14+1 g(7)=15 g(7) = 15 g(7)=15 The value of (g∘h)(−3) (g \circ h)(-3) (g∘h)(−3) is:15 \boxed{15} 15​

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are given a linear function $g(x)$ with selected values in a table.
2. We have the function $h(x) = x^2 - 2$ .
3. We need to evaluate the composite function $(g \circ h)(-3)$ .

1. Evaluate $h(-3)$ using the function $h(x)$ .
2. Use the result from step 1 to find $g(h(-3))$ using the table of values for $g(x)$ .

Substitute $x = -3$ into $h(x)$ :
$h(-3) = (-3)^2 - 2$

Simplify the expression:
$h(-3) = 9 - 2$ $h(-3) = 7$

Now, we need to evaluate $g(h(-3)) = g(7)$ .
Check the table for the value of $g(7)$ .

Now, use the point-slope form to find the equation of the line:
Using point $(-3, -5)$ :
$y + 5 = 2(x + 3)$
Simplify to find $g(x)$ :
$y = 2x + 6 - 5$ $y = 2x + 1$
Thus, $g(x) = 2x + 1$ .

Now substitute $x = 7$ into $g(x)$ :
$g(7) = 2(7) + 1$ $g(7) = 14 + 1$ $g(7) = 15$
The value of $(g \circ h)(-3)$ is:
$\boxed{15}$